So it is basically a closed chase with brick surrounding the flue. That isn't how it would be built today, but very common on most older homes. What you really don't want is naked flue tiles in an open chase with no brick supporting them because they will be prone to serious failures like the tile stack collapsing.
You don't really need to worry about that cavity. They want the brick cores filled and adequate mortar joints in the brick, not dry set(no mortar).
Construct a new crown to seal off the chase and extend the flue tiles by about 6" off the top of the new crown. There is a chance of a cross breeze pulling smoke down the adjacent flue, but there are a few factors that can cause that and I would need a few photos of the whole picture to have a decent idea if that could be an issue.